Constant current, constant power and constant impedance loads

A constant power load varies it's impedance on change of input voltage to keep the power constant. A constant impedance load is simply a load that presents an unchanging impedance, like a resistor. An L-Pad is used to change speaker output level whilst maintaining a constant impedance load to the amplifier.

A good example of a constant power load is a switching regulator. Since this has to maintain it's power into it's load, it must draw the same power from it's source even if the source changes voltage.
This is also an example of a negative impedance because in order to maintain the output power, if the voltage in drops, the current must rise (opposite to a standard resistor where the current and voltage rise/fall with each other)

Here is an example circuit, made from a LT1377 boost switching regulator:

Here is the simulation:

The input voltage V(in) (blue trace) starts off at 4V, and gradually rises up to 10V. We can see the power (red trace) stays constant at ~1W over a change of 6V at the input (it's not perfect as it's meant to represent "real life" and not 100% efficient, but it's pretty close)
We can also see the dynamic negative resistance characteristic (green trace) which is due to the input current falling as the voltage rises. Tt falls from ~300mA to ~120mA over the voltage rise from 4V to 10V - don't be confused by the minus sign, that's just the direction of measurement in LTSpice.
The dynamic resistance slope can be roughly calculated by (4V - 10V) / (300mA - 120mA) = -33.3Ω. Looking it another way, 6V / -33.3Ω = -180mA.

You are right in what constant current, power, and impedance loads are. However, most loads are not so nice and predictable. Some loads may look mostly resistive, like heating elements, but many loads have all kinds of screwy characteristics or vary dynamically. For example, the input to switching power supplies look largely like negative resistances because they are mostly constant power. Sometimes the power supply circuitry is specifically designed to present a certain load profile to the input. An AC to DC power supply with power factor correction is a good example of that.

To determine the characteristics of an unknown load, you have to measure it. Keep in mind though, that many loads don't fall into a nice neat category like constant current, power, or resistance. Think of a microcontroller, for example. It's characteristics can change dramatically and quickly depending on what it is doing inside and how it may be driving output pins.